A306927 a(n) = A001615(n) - n.

0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72

Offset: 1

Author

Torlach Rush, Mar 16 2019

Keywords

• nonn
• easy

Comments

Analogous to A051953.

a(n) = A051953(n) if n is an element of A000961.

a(n) > A051953(n) if n is an element of A024619.

The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019

Examples

			0 is a term because A001615(1) - 1 = 0.
1 is a term because A001615(2) - 2 = 1.
3 is a term because A001615(9) - 9 = 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Wikipedia, Dedekind psi function.

Crossrefs

Cf. A000961, A001615, A024619, A051953, A082020.

Programs

Mathematica

a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)

PARI

a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019

Formula

a(n) = A001615(n) - n.

a(n) = Sum_{d|n, dAmiram Eldar, Sep 06 2019

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023